One Factor Design: A Comprehensive Guide

One Factor Design: An Introduction

A one factor design (also known as a one-way ANOVA) is a statistical method used to determine if there are significant differences between the means of multiple groups. In this design, there is one independent variable (factor) with multiple levels or categories.

Suppose $y_{ij}$ is the response is the $i$th treatment for the $j$th experimental unit, where $i=1,2,\cdots, I$. The statistical model for a completely randomized one-factor design that leads to a One-Way ANOVA is

$$y_{ij} = \mu_i + e_{ij}$$

where $\mu_i$ is the unknown (population) mean for all potential responses to the $i$th treatment, and $e_{ij}$ is the error (deviation of the response from population mean).

The responses within and across treatments are assumed to be independent and normally distributed random variables with constant variance.

One Factor Design’s Statistical Model

Let $\mu = \frac{1}{I} \sum \limits_{i} \mu_i$ be the grand mean or average of the population means. Let $\alpha_i=\mu_i-\mu$ be the $i$th group treatment effect. The treatment effects are constrained to add to zero ($\alpha_1+\alpha_2+\cdots+\alpha_I=0$) and measure the difference between the treatment population means and the grand mean.

Therefore the one way ANOVA model is $$y{ij} = \mu + \alpha_i + e_{ij}$$

$$Response = \text{Grand Mean} + \text{Treatment Effect} + \text{Residuals}$$

From this model, the hypothesis of interest is whether the population means are equal:

$$H_0:\mu_1=\mu_2= \cdots = \mu_I$$

The hypothesis is equivalent to $H_0:\alpha_1 = \alpha_2 =\cdots = \alpha_I=0$. If $H_0$ is true, then the one-way ANOVA model is

$$y_{ij} = \mu + e_{ij}$$ where $\mu$ is the common population mean.

One Factor Design Example

Let’s say you want to compare the average test scores of students from three different teaching methods (Method $A$, Method $B$, and Method $C$).

• Independent variable: Teaching method (with three levels: $A, B, C$)
• Dependent variable: Test scores

When to Use a One Factor Design

• Comparing means of multiple groups: When one wants to determine if there are significant differences in the mean of a dependent variable across different groups or levels of a factor.
• Exploring the effect of a categorical variable: When one wants to investigate how a categorical variable influences a continuous outcome.

Assumptions of One-Factor ANOVA

• Normality: The data within each group should be normally distributed.
• Homogeneity of variance (Equality of Variances): The variances of the populations from which the samples are drawn should be equal.
• Independence: The observations within each group should be independent of each other.

When to Use One Factor Design

• When one wants to compare the means of multiple groups.
• When the independent variable has at least three levels.
• When the dependent variable is continuous (e.g., numerical).

Note that

If The Null hypothesis is rejected, one can perform post-hoc tests (for example, Tukey’s HSD, Bonferroni) to determine which specific groups differ significantly from each other.

Remember: While one-factor designs are useful for comparing multiple groups, they cannot establish causation.

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